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Linearized models in PET. Vesa Oikonen. http://pet.utu.fi/staff/vesoik/modelling/workshop2/linmod.ppt. 2003-06-05 Turku PET Centre – Modelling workshop Modelling workshop. 3-compartment model. K 1. k 3. C PLASMA. C FREE. C BOUND. k 2. k 4.

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Linearized models in pet

Linearized models in PET

Vesa Oikonen

http://pet.utu.fi/staff/vesoik/modelling/workshop2/linmod.ppt

2003-06-05 Turku PET Centre – Modelling workshop

Modelling workshop


3 compartment model
3-compartment model

K1

k3

CPLASMA

CFREE

CBOUND

k2

k4



Nonlinear estimation of model parameters
Nonlinear estimation of model parameters

Iterative minimization of weighted residual sum of squares


Linearization 1
Linearization #1

Sums, substitutions, rearrangements, integration

http://pet.utu.fi/staff/vesoik/reports/tpcmod0007.pdf


Linearization 11
Linearization #1

”Logan” plot

Slope = DV

Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670.


Linearization 2
Linearization #2

Sums, substitutions, rearrangements, two integrations

Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab 1984;4:629-632


Linearization 21
Linearization #2

or after rearrangement

Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...)


Same method applied to simplified reference tissue model srtm
Same method applied to simplified reference tissue model (SRTM)

Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...)

Zhou Y et al. Linear regression with spatial constraint to generate parametric images of ligand-receptor dynamic PET studies with a simplified reference tissue model. NeuroImage 2003;18:975-989.


3 compartment model irreversible binding or trapped metabolite
3-compartment model (SRTM)irreversible binding or trapped metabolite

K1

k3

CPLASMA

CFREE

CBOUND

k2

k4=0


Differential equations for compartment concentrations1
Differential equations for compartment concentrations (SRTM)

K1

k3

CPLASMA

CFREE

CBOUND

k2


Linearization 1 k 4 0
Linearization #1 (SRTM)(k4=0)

Sums, substitutions, rearrangements, integration

http://pet.utu.fi/staff/vesoik/reports/tpcmod0006.pdf


Linearization 1 k 4 01
Linearization #1 (SRTM)(k4=0)

”Patlak” plot

Slope = Ki

Logan J. Graphical analysis of PET data applied to reversible and irreversible tracers. Nucl Med Biol 2000;27:661-670.


Linearization 2 k 4 0
Linearization #2 (SRTM)(k4=0)

Sums, substitutions, rearrangements, two integrations

Blomqvist G. On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab 1984;4:629-632


Linearization 2 k 4 01
Linearization #2 (SRTM)(k4=0)

or after rearrangement

Parameters can be solved by multilinear regression (Y=p1x1+p2x2+...)


Nonlinear models
Nonlinear models (SRTM)

  • Easy to set constraints for parameters

  • Applicable to all compartmental settings

  • Straightforward weighting

  • Predictable noise properties

  • Non-linearly affected by PVE and heterogeneity

  • Slow estimation of parameters

  • Commonly local minima

  • Cannot be applied to sinogram data


Linear models
Linear models (SRTM)

  • Fast calculation

  • Applicable to sinogram data

  • Linearly affected by PVE and heterogeneity

  • All models can not be linearized

  • Constraining parameters may be difficult

  • Noise may lead to bias

  • Weights are difficult to determine


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